update paper

manuscript/Ambarkutuk-Paper-1-Manuscript.aux

@@ -49,34 +49,32 @@
 \@writefile{toc}{\contentsline {subsection}{\numberline {3.4}Sensor Fusion Scheme}{8}\protected@file@percent }
 \newlabel{eq:optimization}{{14}{8}}
 \newlabel{eq:optimization@cref}{{[equation][14][]14}{[1][8][]8}}
-\newlabel{eq:final}{{15}{9}}
-\newlabel{eq:final@cref}{{[equation][15][]15}{[1][9][]9}}
-\newlabel{sec:validation}{{4}{10}}
-\newlabel{sec:validation@cref}{{[section][4][]4}{[1][10][]10}}
-\@writefile{toc}{\contentsline {section}{\numberline {4}Validation}{10}\protected@file@percent }
-\@writefile{tdo}{\contentsline {todo}{These sections are commented out for now.}{10}\protected@file@percent }
+\newlabel{sec:validation}{{4}{9}}
+\newlabel{sec:validation@cref}{{[section][4][]4}{[1][9][]9}}
+\@writefile{toc}{\contentsline {section}{\numberline {4}Validation}{9}\protected@file@percent }
+\@writefile{tdo}{\contentsline {todo}{These sections are commented out for now.}{9}\protected@file@percent }
 \pgfsyspdfmark {pgfid6}{7292190}{42843472}
 \pgfsyspdfmark {pgfid9}{34931998}{42723869}
 \pgfsyspdfmark {pgfid10}{36357406}{42589520}
-\newlabel{sec:conclusion}{{5}{10}}
-\newlabel{sec:conclusion@cref}{{[section][5][]5}{[1][10][]10}}
-\@writefile{toc}{\contentsline {section}{\numberline {5}Conclusions \& Future Work}{10}\protected@file@percent }
+\newlabel{sec:conclusion}{{5}{9}}
+\newlabel{sec:conclusion@cref}{{[section][5][]5}{[1][9][]9}}
+\@writefile{toc}{\contentsline {section}{\numberline {5}Conclusions \& Future Work}{9}\protected@file@percent }
 \@writefile{toc}{\let\numberline\tmptocnumberline}
-\@writefile{toc}{\contentsline {section}{\numberline {Appendix \nobreakspace  {}A}Theorems}{10}\protected@file@percent }
-\newlabel{thm:rayleigh-deterministic}{{2}{10}}
-\newlabel{thm:rayleigh-deterministic@cref}{{[thm][2][2147483647]2}{[1][10][]10}}
-\newlabel{thm:rayleigh-stochastic}{{3}{10}}
-\newlabel{thm:rayleigh-stochastic@cref}{{[thm][3][2147483647]3}{[1][10][]10}}
-\newlabel{eq:rayleigh_prob}{{A.3}{10}}
-\newlabel{eq:rayleigh_prob@cref}{{[equation][3][2147483647,1]A.3}{[1][10][]10}}
-\newlabel{thm:noncentral-chi}{{4}{11}}
-\newlabel{thm:noncentral-chi@cref}{{[thm][4][2147483647]4}{[1][10][]11}}
-\newlabel{thm:normal-approximation}{{5}{11}}
-\newlabel{thm:normal-approximation@cref}{{[thm][5][2147483647]5}{[1][11][]11}}
-\newlabel{thm:lotus}{{6}{11}}
-\newlabel{thm:lotus@cref}{{[thm][6][2147483647]6}{[1][11][]11}}
-\newlabel{thm:density}{{7}{12}}
-\newlabel{thm:density@cref}{{[thm][7][2147483647]7}{[1][12][]12}}
+\@writefile{toc}{\contentsline {section}{\numberline {Appendix \nobreakspace  {}A}Theorems}{9}\protected@file@percent }
+\newlabel{thm:rayleigh-deterministic}{{2}{9}}
+\newlabel{thm:rayleigh-deterministic@cref}{{[thm][2][2147483647]2}{[1][9][]9}}
+\newlabel{thm:rayleigh-stochastic}{{3}{9}}
+\newlabel{thm:rayleigh-stochastic@cref}{{[thm][3][2147483647]3}{[1][9][]9}}
+\newlabel{eq:rayleigh_prob}{{A.3}{9}}
+\newlabel{eq:rayleigh_prob@cref}{{[equation][3][2147483647,1]A.3}{[1][9][]9}}
+\newlabel{thm:noncentral-chi}{{4}{10}}
+\newlabel{thm:noncentral-chi@cref}{{[thm][4][2147483647]4}{[1][9][]10}}
+\newlabel{thm:normal-approximation}{{5}{10}}
+\newlabel{thm:normal-approximation@cref}{{[thm][5][2147483647]5}{[1][10][]10}}
+\newlabel{thm:lotus}{{6}{10}}
+\newlabel{thm:lotus@cref}{{[thm][6][2147483647]6}{[1][10][]10}}
+\newlabel{thm:density}{{7}{11}}
+\newlabel{thm:density@cref}{{[thm][7][2147483647]7}{[1][11][]11}}
 \bibstyle{elsarticle-num-names}
 \bibdata{etc/cas-refs}
 \bibcite{covo2014novel}{{1}{2014}{{Covo et~al.}}{{Covo, Elalouf et~al.}}}
@@ -87,7 +85,7 @@
 \bibcite{hartley2003multiple}{{6}{2003}{{Hartley and Zisserman}}{{}}}
 \bibcite{alajlouni2019new}{{7}{2019}{{Alajlouni and Tarazaga}}{{}}}
 \bibcite{bahroun2014new}{{8}{2014}{{Bahroun et~al.}}{{Bahroun, Michel, Frassati, Carmona, and Lacoume}}}
-\@writefile{toc}{\contentsline {section}{References}{14}\protected@file@percent }
+\@writefile{toc}{\contentsline {section}{References}{13}\protected@file@percent }
 \bibcite{poston2016indoor}{{9}{2016}{{Poston et~al.}}{{Poston, Buehrer, Woolard, and Tarazaga}}}
 \bibcite{poston2017indoor}{{10}{2017}{{Poston et~al.}}{{Poston, Buehrer, and Tarazaga}}}
 \bibcite{kessler2019vibration}{{11}{2019{}}{{Kessler et~al.}}{{Kessler, Malladi, and Tarazaga}}}

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  []
 
-[12]) [13] (./Ambarkutuk-Paper-1-Manuscript.bbl [14
+[11]) [12] (./Ambarkutuk-Paper-1-Manuscript.bbl [13
 
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manuscript/sections/technique_new.tex

@@ -51,7 +51,7 @@ Without losing the generality, we model the measurement vector $\vect{z}_i$ as t
 where $\vect{z}_{true, i} = \left(z_i[1],\ldots, z_i[n]\right)^\top$ and $\vect{\zeta}_i = \left(\zeta_i[1], \ldots, \zeta_i[n]\right)^\top$.
 It is quite common to see the disturbance vector $\vect{\zeta}_i$ as Normally distributed \texttt{iid}\footnote{Indepedently and identically distributed} random variables, i.e., $\zeta_i[k] \sim \N{\mu_\zeta}{\sigma_\zeta}$.
 In this study, we found that the signal energy $e_i \sim \f{E_i} = \chi^{\prime, 2}_n \left( \lambda \right)$ exactly follows a Non-central Chi-squared Distrubtion with $n$ degrees-of-freedom (number of samples that the energy is calculated with) and noncentrality parameter $\lambda$, (c.f. \Cref{thm:noncentral-chi}).
-With this, we propose our first theorem showing $\f{E_i}$ can be safely approximated with a Normally distributed random variable.
+With this, we present our first theorem showing $\f{E_i}$ can be safely approximated with a Normally distributed random variable.
 
 \begin{thm}[Energy distribution of a stochastic representation of accelerometer measurements converges safely Normal distribution.]
     \label{thm:opus-theorem}
@@ -150,7 +150,7 @@ As all the sensors' are independent from each other, we can represent the joint-
 \end{equation}
 
 Let $\kappa_i$ be the our definition of \gls{snr} where it is calculated as $\kappa_i = \dfrac{e_{true,i}}{e_i}$.
-Assuming all the sensors have somewhat close \gls{snr} value, we can define a shared \gls{snr} value for all sensors $\kappa = \kappa_1 \approxeq  \ldots \approxeq \kappa_m$.
+Assuming all the sensors have somewhat close \gls{snr} value, we can define a shared \gls{snr} values for all sensors $\kappa = \kappa_1 \approxeq  \ldots \approxeq \kappa_m$.
 Given this definition, we can reparameterize the joint-\gls{pdf} as given below:
 
 \begin{equation}
@@ -160,28 +160,21 @@ Given this definition, we can reparameterize the joint-\gls{pdf} as given below:
 Given this representation we can define a cost function $J(\kappa; \vect{x})$ to minimize as a function $0\leq\kappa\leq1$: 
 
 \begin{equation}
-    J\left(\kappa; \forall \vect{x} \in \mathcal{S} \right) = -\sum_{\vect{x}}^{\forall \vect{x} \in \mathcal{S}}\sum_{i=1}^{m} \log{f_{\vect{X}_i}\left(\vect{x}; \kappa, \vect{\beta}\right)} 
+    J\left(\kappa\right) = -\oint_{\vect{x} \in \mathcal{S}}\sum_{i=1}^{m} \log{f_{\vect{X}_i}\left(\vect{x}; \kappa, \vect{\beta}\right)} \, \diff{\vect{x}}
 \end{equation}
 
 The optimization problem is then in hand can be stated as given in \Cref{eq:optimization}.
 \begin{equation}
     \label{eq:optimization}
-    \kappa^* = \argmin_{\kappa} J\left(\kappa, \cdot \right)
+    \kappa^* = \argmin_{\kappa} J\left(\kappa\right)
 \end{equation}
-By the virtue of optimality, the solution can be obtained by solving the equation $\dfrac{\diff J}{\diff\kappa} = 0$ which yields the solution $\kappa^*$:
-\begin{equation*}
-    \kappa^* = \frac{1}{m} \sum_{i=1}^{m}\left\lvert \frac{\frac{e_i\,\sqrt{n^2\,{\sigma_{\zeta}}^4-4\,n\,{\sigma_{\zeta}}^2\,g^{-1}\left(\norm{\vect{x}_i - \vect{x}_{a_i}}_2\right)+16\,{\sigma_{\zeta}}^4+4\,{g^{-1}\left(\norm{\vect{x}_i - \vect{x}_{a_i}}_2\right)}^2}}{2}+\frac{e_i\,{\sigma_{\zeta}}^2\,\left(n+4\right)}{2}}{{g^{-1}\left(\norm{\vect{x}_i - \vect{x}_{a_i}}_2\right)}^2-n\,\left(2\,{\sigma_{\zeta}}^4+g^{-1}\left(\norm{\vect{x}_i - \vect{x}_{a_i}}_2\right)\,{\sigma_{\zeta}}^2\right)} \right\rvert
-\end{equation*}
-
-Finally, the result of impact location can be obtained by plugging $\kappa^*$ in the joint-\gls{pdf}.
-\begin{equation}
-    \label{eq:final}
-    \vect{x}^* = \argmax_{\vect{x}} \prod_{i=1}^{m}f_{\vect{X}_i}\left(\vect{x}; \kappa^*, \vect{\beta}\right)
-\end{equation}
-
+% By the virtue of optimality, the solution can be obtained by solving the equation $\dfrac{\diff J}{\diff\kappa} = 0$ which yields the solution $\kappa^*$:
 % \begin{equation*}
-%     \left\lvert \frac{a^{2/b}\,e\,\sqrt{{\left(\frac{\norm{\vect{x}_i - \vect{x}_{a_i}}_2}{a}\right)}^{2/b}+4\,{\sigma_{\zeta}}^4+\frac{n^2\,{\sigma_{\zeta}}^4}{4}-\frac{n\,{\sigma_{\zeta}}^2\,{\left(\norm{\vect{x}_i - \vect{x}_{a_i}}\right)}^{\frac{1}{b}}}{a^{1/b}}}-\frac{a^{2/b}\,e\,{\sigma_{\zeta}}^2\,\left(n+4\right)}{2}}{2\,a^{2/b}\,n\,{\sigma_{\zeta}}^4-{\left(\norm{\vect{x}_i - \vect{x}_{a_i}}\right)}^{2/b}+a^{1/b}\,n\,{\sigma_{\zeta}}^2\,{\left(\norm{\vect{x}_i - \vect{x}_{a_i}}\right)}^{\frac{1}{b}}} \right\rvert
+%     \kappa^* = \frac{1}{m} \sum_{i=1}^{m}\left\lvert \frac{\frac{e_i\,\sqrt{n^2\,{\sigma_{\zeta}}^4-4\,n\,{\sigma_{\zeta}}^2\,g^{-1}\left(\norm{\vect{x}_i - \vect{x}_{a_i}}_2\right)+16\,{\sigma_{\zeta}}^4+4\,{g^{-1}\left(\norm{\vect{x}_i - \vect{x}_{a_i}}_2\right)}^2}}{2}+\frac{e_i\,{\sigma_{\zeta}}^2\,\left(n+4\right)}{2}}{{g^{-1}\left(\norm{\vect{x}_i - \vect{x}_{a_i}}_2\right)}^2-n\,\left(2\,{\sigma_{\zeta}}^4+g^{-1}\left(\norm{\vect{x}_i - \vect{x}_{a_i}}_2\right)\,{\sigma_{\zeta}}^2\right)} \right\rvert
 % \end{equation*}
-\begin{equation*}
-    \left(\begin{array}{c} \frac{e_{i}\,\left|n\,{\sigma _{i}}^2+\sqrt{n^2\,{\sigma _{i}}^4-4\,n\,{\sigma _{i}}^2\,g_{\mathrm{inv}}\left(\sqrt{x^2-2\,x\,x_{i}+{x_{i}}^2+y^2-2\,y\,y_{i}+{y_{i}}^2}\right)+16\,{\sigma _{i}}^4+4\,{g_{\mathrm{inv}}\left(\sqrt{x^2-2\,x\,x_{i}+{x_{i}}^2+y^2-2\,y\,y_{i}+{y_{i}}^2}\right)}^2}+4\,{\sigma _{i}}^2\right|}{2\,\left|2\,n\,{\sigma _{i}}^4+n\,{\sigma _{i}}^2\,g_{\mathrm{inv}}\left(\sqrt{x^2-2\,x\,x_{i}+{x_{i}}^2+y^2-2\,y\,y_{i}+{y_{i}}^2}\right)-{g_{\mathrm{inv}}\left(\sqrt{x^2-2\,x\,x_{i}+{x_{i}}^2+y^2-2\,y\,y_{i}+{y_{i}}^2}\right)}^2\right|}\\ \frac{e_{i}\,\left|n\,{\sigma _{i}}^2-\sqrt{n^2\,{\sigma _{i}}^4-4\,n\,{\sigma _{i}}^2\,g_{\mathrm{inv}}\left(\sqrt{x^2-2\,x\,x_{i}+{x_{i}}^2+y^2-2\,y\,y_{i}+{y_{i}}^2}\right)+16\,{\sigma _{i}}^4+4\,{g_{\mathrm{inv}}\left(\sqrt{x^2-2\,x\,x_{i}+{x_{i}}^2+y^2-2\,y\,y_{i}+{y_{i}}^2}\right)}^2}+4\,{\sigma _{i}}^2\right|}{2\,\left|2\,n\,{\sigma _{i}}^4+n\,{\sigma _{i}}^2\,g_{\mathrm{inv}}\left(\sqrt{x^2-2\,x\,x_{i}+{x_{i}}^2+y^2-2\,y\,y_{i}+{y_{i}}^2}\right)-{g_{\mathrm{inv}}\left(\sqrt{x^2-2\,x\,x_{i}+{x_{i}}^2+y^2-2\,y\,y_{i}+{y_{i}}^2}\right)}^2\right|} \end{array}\right)
-\end{equation*}
+
+% Finally, the result of impact location can be obtained by plugging $\kappa^*$ in the joint-\gls{pdf}.
+% \begin{equation}
+%     \label{eq:final}
+%     \vect{x}^* = \argmax_{\vect{x}} \prod_{i=1}^{m}f_{\vect{X}_i}\left(\vect{x}; \kappa^*, \vect{\beta}\right)
+% \end{equation}